Subgroups — #34204191

Order 20224.

478 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098517	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×(C₂×D₇₉)⋊C₁₆	C₂	1	2	yes	maximal
4	#23761477	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11929607	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₇₉	C₂²	1	1	yes	characteristic
8	C₂×#14579118	C₄	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
10	#19098517	—	2	2	no	—
11	#19098517	—	2	2	no	—
12	C₂×#14579072	C₄	1	1	yes	characteristic
13	C₂×#14579071	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
16	(C₂×D₇₉)⋊C₁₆	—	2	4	no	—
17	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₇₉	—	2	2	no	—
20	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂×#11929607	—	2	2	no	—
22	C₂×#11929607	—	2	2	no	—
23	C₂×#11929607	—	2	2	no	—
24	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₇₉	—	2	1	no	—
27	C₂×C₈×D₇₉	—	2	1	no	—
28	C₂×C₈×D₇₉	—	2	1	no	—
29	C₂×C₈×D₇₉	—	2	1	no	—
30	C₂×C₈×D₇₉	D₄	1	2	yes	—
31	C₂×C₈×D₇₉	Q₈	1	2	yes	—
32	#14579118	—	2	2	no	—
33	#14579118	—	2	2	no	—
34	C₂×Dic₇₉⋊C₄	C₈	1	1	yes	characteristic
35	C₂×Dic₇₉⋊C₄	C₈	1	1	yes	characteristic
36	C₂×(C₂×D₇₉)⋊C₄	C₈	1	1	yes	characteristic
37	C₂×(C₂×D₇₉)⋊C₄	C₈	1	1	yes	characteristic
38	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₇₉⋊C₁₆	—	2	2	no	—
40	C₂×C₇₉⋊C₁₆	—	2	2	no	—
41	C₂×C₇₉⋊C₁₆	—	2	2	no	—
42	#14579072	—	2	2	no	—
43	#14579071	—	2	2	no	—
44	C₈×Dic₇₉	—	2	1	no	—
45	C₈×Dic₇₉	—	2	1	no	—
46	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₇₉	—	2	2	no	—
50	C₂×C₁₆×C₇₉	—	2	2	no	—
51	C₂×C₁₆×C₇₉	—	2	2	no	—
52	C₄×C₈×C₇₉	D₄	1	2	yes	—
53	C₄×C₈×C₇₉	D₄	1	2	yes	—
54	C₂³×D₇₉	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₇₉	C₂×C₈	1	1	yes	characteristic
56	C₂²×Dic₇₉	C₂×C₈	1	1	yes	characteristic
57	C₂²×Dic₇₉	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄×D₇₉	—	2	1	no	—
59	C₂×C₄×D₇₉	—	2	1	no	—
60	C₂×C₄×D₇₉	—	2	1	no	—
61	C₂×C₄×D₇₉	—	2	1	no	—
62	C₂×C₄×D₇₉	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×D₇₉	M₁₆	1	2	yes	—
64	#11929607	—	2	8	no	—
65	C₂×C₇₉⋊C₈	—	2	1	no	—
66	C₂×C₇₉⋊C₈	—	2	1	no	—
67	C₂×C₇₉⋊C₈	—	2	1	no	—
68	C₂×C₇₉⋊C₈	—	2	1	no	—
69	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₇₉⋊C₈	M₁₆	1	2	yes	—
71	C₈×D₇₉	—	2	2	no	—
72	C₈×D₇₉	—	2	2	no	—
73	C₈×D₇₉	—	2	2	no	—
74	C₈×D₇₉	—	2	2	no	—
75	Dic₇₉⋊C₄	—	2	2	no	—
76	Dic₇₉⋊C₄	—	2	2	no	—
77	(C₂×D₇₉)⋊C₄	—	2	2	no	—
78	(C₂×D₇₉)⋊C₄	—	2	2	no	—
79	C₄×Dic₇₉	—	2	1	no	—
80	C₄×Dic₇₉	—	2	1	no	—
81	C₇₉⋊C₁₆	—	4	2	no	—
82	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	—	2	1	no	—
86	C₂×C₈×C₇₉	—	2	1	no	—
87	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
93	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
94	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
95	C₄²×C₇₉	M₁₆	1	2	yes	—
96	C₁₆×C₇₉	—	4	2	no	—
97	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
98	C₂²×D₇₉	—	2	1	no	—
99	C₂²×D₇₉	—	2	1	no	—
100	C₂²×D₇₉	—	2	2	no	—
101	C₂²×D₇₉	—	2	2	no	—
102	C₂²×D₇₉	C₄⋊C₈	1	2	yes	—
103	C₂×C₄×C₇₉	—	2	2	no	—
104	C₂×C₄×C₇₉	—	2	2	no	—
105	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
113	C₂×Dic₇₉	—	2	1	no	—
114	C₂×Dic₇₉	—	2	1	no	—
115	C₂×Dic₇₉	—	2	1	no	—
116	C₂×Dic₇₉	—	2	1	no	—
117	C₂×Dic₇₉	—	2	2	no	—
118	C₂×Dic₇₉	—	2	2	no	—
119	C₂×Dic₇₉	C₄⋊C₈	1	2	yes	—
120	C₄×D₇₉	—	2	2	no	—
121	C₄×D₇₉	—	2	2	no	—
122	C₄×D₇₉	—	2	2	no	—
123	C₄×D₇₉	—	2	2	no	—
124	C₈×C₇₉	—	2	2	no	—
125	C₈×C₇₉	#121	1	1	yes	characteristic
126	C₈×C₇₉	#121	1	1	yes	characteristic
127	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
129	C₇₉⋊C₈	—	2	2	no	—
130	C₇₉⋊C₈	—	2	2	no	—
131	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
133	C₂²×C₇₉	#431	1	1	yes	characteristic
134	C₂²×C₇₉	#504	1	2	yes	—
135	C₂²×C₇₉	#504	1	2	yes	—
136	C₂×D₇₉	—	2	2	no	—
137	C₂×D₇₉	—	2	4	no	—
138	C₂×D₇₉	—	4	1	no	—
139	C₂×D₇₉	—	4	1	no	—
140	C₂×D₇₉	—	4	2	no	—
141	C₄×C₇₉	—	2	2	no	—
142	C₄×C₇₉	—	2	2	no	—
143	C₄×C₇₉	#380	1	1	yes	characteristic
144	C₄×C₇₉	#407	1	1	yes	characteristic
145	C₄×C₇₉	#423	1	1	yes	characteristic
146	C₄×C₇₉	#432	1	1	yes	characteristic
147	Dic₇₉	—	2	2	no	—
148	Dic₇₉	—	4	1	no	—
149	Dic₇₉	—	4	1	no	—
150	#12784	—	79	1	no	maximal, Sylow
151	C₂×C₇₉	#1797	1	1	yes	characteristic
152	C₂×C₇₉	#2113	1	1	yes	characteristic
153	C₂×C₇₉	#2113	1	1	yes	characteristic
154	C₂×C₇₉	#2151	1	2	yes	—
155	C₂×C₇₉	#2164	1	2	yes	—
156	D₇₉	—	4	2	no	—
157	C₂×#405	—	79	1	no	—
158	C₂×C₄⋊C₁₆	—	79	1	no	—
159	C₂×C₄⋊C₁₆	D₇₉	1	1	yes	characteristic, complemented
160	C₂×C₂²⋊C₁₆	—	79	2	no	—
161	#1374	—	79	2	no	—
162	C₇₉	#12784	1	1	yes	characteristic, complemented, Sylow
163	C₂²×M₁₆	—	79	1	no	—
164	C₂³×C₈	—	79	1	no	—
165	C₂×#117	—	79	1	no	—
166	C₂²×C₁₆	—	79	2	no	—
167	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
168	#405	—	158	2	no	—
169	#405	—	158	2	no	—
170	C₂×#97	—	79	1	no	—
171	C₂×#97	—	79	1	no	—
172	C₂×C₄×C₈	—	79	1	no	—
173	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
174	C₄⋊C₁₆	—	2	2	no	—
175	C₄⋊C₁₆	—	158	2	no	—
176	C₂²⋊C₁₆	—	158	4	no	—
177	C₂³×C₄	—	79	1	no	—
178	C₂×M₁₆	—	158	2	no	—
179	C₂×M₁₆	—	158	2	no	—
180	C₂×M₁₆	—	158	2	no	—
181	C₂²×C₈	—	79	1	no	—
182	C₂²×C₈	—	79	1	no	—
183	C₂²×C₈	—	79	2	no	—
184	C₂²×C₈	—	79	2	no	—
185	C₂²×C₈	—	158	1	no	—
186	C₂²×C₈	—	158	1	no	—
187	C₂²×C₈	—	158	1	no	—
188	C₂²×C₈	—	158	1	no	—
189	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
191	#117	—	158	2	no	—
192	#117	—	158	2	no	—
193	C₂×C₄⋊C₄	—	79	1	no	—
194	C₂×C₄⋊C₄	—	79	1	no	—
195	C₂×C₂²⋊C₄	—	79	1	no	—
196	C₂×C₂²⋊C₄	—	79	1	no	—
197	C₂×C₄²	—	79	1	no	—
198	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	158	2	no	—
203	C₂×C₁₆	—	158	2	no	—
204	C₂×C₁₆	—	158	2	no	—
205	#97	—	158	2	no	—
206	#97	—	158	2	no	—
207	C₄×C₈	—	158	1	no	—
208	C₄×C₈	—	158	1	no	—
209	C₄×C₈	D₃₁₆	1	2	yes	—
210	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
211	C₂⁴	—	79	1	no	—
212	C₂²×C₄	—	79	1	no	—
213	C₂²×C₄	—	79	2	no	—
214	C₂²×C₄	—	79	2	no	—
215	C₂²×C₄	—	79	1	no	—
216	C₂²×C₄	—	79	1	no	—
217	C₂²×C₄	—	158	1	no	—
218	C₂²×C₄	—	158	1	no	—
219	C₂²×C₄	—	158	1	no	—
220	C₂²×C₄	—	158	1	no	—
221	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
222	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
224	M₁₆	—	158	8	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	79	2	no	—
228	C₂×C₈	—	79	2	no	—
229	C₂×C₈	—	158	1	no	—
230	C₂×C₈	—	158	1	no	—
231	C₂×C₈	—	158	1	no	—
232	C₂×C₈	—	158	1	no	—
233	C₂×C₈	—	158	2	no	—
234	C₂×C₈	—	158	2	no	—
235	C₂×C₈	—	158	2	no	—
236	C₂×C₈	—	158	2	no	—
237	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
238	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
239	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
240	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
241	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
242	C₂×C₈	#11929623	1	1	yes	characteristic
243	C₂×C₈	#11929625	1	1	yes	characteristic
244	C₄⋊C₄	—	158	2	no	—
245	C₄⋊C₄	—	158	2	no	—
246	C₂²⋊C₄	—	158	2	no	—
247	C₂²⋊C₄	—	158	2	no	—
248	C₄²	—	158	1	no	—
249	C₄²	—	158	1	no	—
250	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
251	C₄²	#11929607	1	2	yes	—
252	C₁₆	—	4	2	no	—
253	C₁₆	—	316	2	no	—
254	C₂³	—	79	2	no	—
255	C₂³	—	158	2	no	—
256	C₂³	—	158	2	no	—
257	C₂³	—	158	1	no	—
258	C₂³	—	158	1	no	—
259	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	79	2	no	—
263	C₂×C₄	—	158	2	no	—
264	C₂×C₄	—	158	2	no	—
265	C₂×C₄	—	158	1	no	—
266	C₂×C₄	—	158	1	no	—
267	C₂×C₄	—	158	1	no	—
268	C₂×C₄	—	158	1	no	—
269	C₂×C₄	—	158	2	no	—
270	C₂×C₄	—	158	2	no	—
271	C₂×C₄	—	158	2	no	—
272	C₂×C₄	—	158	2	no	—
273	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
274	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
275	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
277	C₂×C₄	#14579119	1	1	yes	characteristic
278	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
279	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
280	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	158	2	no	—
283	C₈	—	158	2	no	—
284	C₈	#14579132	1	1	yes	characteristic
285	C₈	#14579136	1	1	yes	characteristic
286	C₈	#14579137	1	1	yes	characteristic
287	C₈	#14579141	1	1	yes	characteristic
288	C₂²	—	316	1	no	—
289	C₂²	—	316	1	no	—
290	C₂²	—	316	2	no	—
291	C₂²	—	158	4	no	—
292	C₂²	—	158	2	no	—
293	C₂²	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
294	C₂²	C₄⋊C₈×D₇₉	1	1	yes	characteristic
295	C₂²	#19098615	1	1	yes	characteristic
296	C₂²	#19098979	1	2	yes	—
297	C₂²	#19098981	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	316	1	no	—
301	C₄	—	316	1	no	—
302	C₄	—	158	2	no	—
303	C₄	#19098420	1	1	yes	characteristic
304	C₄	#19098526	1	1	yes	characteristic
305	C₄	#19098585	1	1	yes	characteristic
306	C₄	#19098619	1	1	yes	characteristic
307	C₂	—	316	2	no	—
308	C₂	#23763377	1	1	yes	characteristic
309	C₂	#23765013	1	1	yes	characteristic
310	C₂	#23765015	1	1	yes	characteristic
311	C₂	#23765157	1	2	yes	—
312	C₂	#23765206	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator, socle	C₂³×C₇₉	C₂²×C₈
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
prime radical	C₂³×D₇₉	C₂×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#12784
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	semidirect
4. #23761477	C₂	semidirect
5. C₂×C₄⋊C₁₆×C₇₉	C₂	semidirect
159. C₂×C₄⋊C₁₆	D₇₉	semidirect
162. C₇₉	#12784	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19098517	2
C₂×C₄⋊(C₇₉⋊C₁₆)	2
C₂×(C₂×D₇₉)⋊C₁₆	2
#23761477	2
C₂×C₄⋊C₁₆×C₇₉	2
#12784	79